Branching Brownian motion with rank-based selection and reaction-diffusion equations
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909017699254272 |
|---|---|
| author | Mercer, Jacob |
| author_facet | Mercer, Jacob |
| contents | We consider a family of branching-selection particle systems in which particles branch at time dependent rate $r$ and are killed with a probability which is dependent on their rank via some function $ψ$. We show that, under fairly minimal conditions, the hydrodynamic limit of such a system is given by the reaction-diffusion equation $U_t = \frac12 U_{xx} + r(t)G(U)$ with nonlinearity $G(U)$ which is a function of $ψ$. This is a significant generalisation of the well-studied $N$-BBM process, and is similar to the family of `$(b,D)$-BBM' processes described by Groisman \& Soprano-Loto (arXiv:2008.09460). On the one hand, this allows us to understand common reaction-diffusion equations as limits of interacting particle systems with simple descriptions. On the other hand, the asymptotic behaviour of solutions of the reaction-diffusion PDEs can help us predict the asymptotic properties of the associated particle systems. We give general conditions under which the branching-selection particle system has an asymptotic velocity, and describe the velocity up to order $(\log N)^{-2}$; furthermore, we describe the connection between this velocity and the spreading speeds and travelling waves of the corresponding reaction-diffusion equation. This provides a partial weak selection principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04860 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Branching Brownian motion with rank-based selection and reaction-diffusion equations Mercer, Jacob Probability 60J70 (Primary) 35K57 (Secondary) We consider a family of branching-selection particle systems in which particles branch at time dependent rate $r$ and are killed with a probability which is dependent on their rank via some function $ψ$. We show that, under fairly minimal conditions, the hydrodynamic limit of such a system is given by the reaction-diffusion equation $U_t = \frac12 U_{xx} + r(t)G(U)$ with nonlinearity $G(U)$ which is a function of $ψ$. This is a significant generalisation of the well-studied $N$-BBM process, and is similar to the family of `$(b,D)$-BBM' processes described by Groisman \& Soprano-Loto (arXiv:2008.09460). On the one hand, this allows us to understand common reaction-diffusion equations as limits of interacting particle systems with simple descriptions. On the other hand, the asymptotic behaviour of solutions of the reaction-diffusion PDEs can help us predict the asymptotic properties of the associated particle systems. We give general conditions under which the branching-selection particle system has an asymptotic velocity, and describe the velocity up to order $(\log N)^{-2}$; furthermore, we describe the connection between this velocity and the spreading speeds and travelling waves of the corresponding reaction-diffusion equation. This provides a partial weak selection principle. |
| title | Branching Brownian motion with rank-based selection and reaction-diffusion equations |
| topic | Probability 60J70 (Primary) 35K57 (Secondary) |
| url | https://arxiv.org/abs/2605.04860 |